Editing Definition (section)

==In logic, mathematics and computing==
In mathematics, definitions are generally not used to describe existing terms, but to describe or characterize a concept.<ref>David Hunter (2010) Essentials of Discrete Mathematics. Jones & Bartlett Publishers, Section 14.1</ref> For naming the object of a definition mathematicians can use either a [[neologism]] (this was mainly the case in the past) or words or phrases of the common language (this is generally the case in modern mathematics). The precise meaning of a term given by a mathematical definition is often different from the English definition of the word used,<ref>Kevin Houston (2009) How to Think Like a Mathematician: A Companion to Undergraduate Mathematics. Cambridge University Press, p. 104</ref> which can lead to confusion, particularly when the meanings are close. For example, a [[set (mathematics)|set]] is not exactly the same thing in mathematics and in common language. In some case, the word used can be misleading; for example, a [[real number]] has nothing more (or less) real than an [[imaginary number]].  Frequently, a definition uses a phrase built with common English words, which has no meaning outside mathematics, such as [[primitive group]] or [[irreducible variety]].

In first-order logic definitions are usually introduced using [[extension by definition]] (so using a metalogic). On the other hand, [[lambda-calculi]] are a kind of logic where the definitions are included as the feature of the formal system itself.

===Classification===
Authors have used different terms to classify definitions used in formal languages like mathematics.  [[Norman Swartz]] classifies a definition as "stipulative" if it is intended to guide a specific discussion.   A stipulative definition might be considered a temporary, working definition, and can only be disproved by showing a logical contradiction.<ref>{{cite web|url=https://www.sfu.ca/philosophy/swartz/definitions.htm#part5.1|title=Norman Swartz - Biography|work=sfu.ca}}</ref> In contrast, a "descriptive" definition can be shown to be "right" or "wrong" with reference to general usage.

Swartz defines a ''[[precising definition]]'' as one that extends the descriptive dictionary definition (lexical definition) for a specific purpose by including additional criteria. A precising definition narrows the set of things that meet the definition.

[[Charles Stevenson (philosopher)|C.L. Stevenson]] has identified ''[[persuasive definition]]'' as a form of stipulative definition which purports to state the "true" or "commonly accepted" meaning of a term, while in reality stipulating an altered use (perhaps as an argument for some specific belief). Stevenson has also noted that some definitions are "legal" or "coercive"&nbsp;– their object is to create or alter rights, duties, or crimes.<ref>Stevenson, C.L., ''Ethics and Language'', Connecticut 1944</ref>

===Recursive definitions===
A [[recursive definition]], sometimes also called an ''inductive'' definition, is one that defines a word in terms of itself, so to speak, albeit in a useful way. Normally this consists of three steps:
# At least one thing is stated to be a member of the set being defined; this is sometimes called a "base set".
# All things bearing a certain relation to other members of the set are also to count as members of the set. It is this step that makes the definition [[Recursion|recursive]].
# All other things are excluded from the set

For instance, we could define a [[natural number]] as follows (after [[Peano axioms|Peano]]): 
# "0" is a natural number.
# Each natural number has a unique successor, such that:
#* the successor of a natural number is also a natural number;
#* distinct natural numbers have distinct successors;
#* no natural number is succeeded by "0".
# Nothing else is a natural number.

So "0" will have exactly one successor, which for convenience can be called "1". In turn, "1" will have exactly one successor, which could be called "2", and so on. The second condition in the definition itself refers to natural numbers, and hence involves [[self-reference]]. Although this sort of definition involves a form of [[Circular definition|circularity]], it is not [[Vicious circle principle|vicious]], and the definition has been quite successful.

In the same way, we can define [[ancestor]] as follows:
#A parent is an ancestor.
#A parent of an ancestor is an ancestor.
#Nothing else is an ancestor.
Or simply: an ancestor is a parent or a parent of an ancestor.